Sharp bounds for symmetric and asymmetric Diophantine approximation

TL;DR

This paper improves bounds on the quality of rational approximations via continued fractions using geometric methods, providing sharper bounds and analyzing their frequency of occurrence.

Contribution

It introduces a geometric approach to refine bounds for continued fraction approximations and determines their asymptotic frequency.

Findings

Sharpened bounds for approximation quality.

Derived sharp upper and lower bounds.

Calculated asymptotic frequency of bounds occurrence.

Abstract

In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur.